Summary: ON THE LOCAL QUOTIENT STRUCTURE OF ARTIN STACKS
JAROD ALPER
ABSTRACT. We show that near closed points with linearly reductive stabilizer, Artin stacks are for-
mally locally quotient stacks by the stabilizer. We conjecture that the statement holds ´etale locally
and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a
point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space, gen-
eralizing the results of Pinkham and Rim. We provide a generalization and stack-theoretic proof of
Luna's ´etale slice theorem which shows that GIT quotient stacks are ´etale locally quotients stacks by
the stabilizer.
1. INTRODUCTION
This paper is motivated by the question of whether an Artin stack is "locally" near a point a
quotient stack by the stabilizer at that point. While this question may appear quite technical in
nature, we hope that a positive answer would lead to intrinsic constructions of moduli schemes
parameterizing objects with infinite automorphisms (e.g. vector bundles on a curve) without the
use of classical geometric invariant theory.
We restrict ourselves to studying Artin stacks X over a base S near closed points |X| with
linearly reductive stabilizer.
We conjecture that this question has an affirmative answer in the ´etale topology. Precisely,
Conjecture 1. If X is an Artin stack finitely presented over an algebraic space S and |X| is a
closed point with linearly reductive stabilizer with image s S, then there exists an ´etale neighborhood